In the context of Navier–Stokes–Cahn–Hilliard models for two-phase flows, in order to overcome the aforementioned limitations (REF ) and (REF ),
the authors in [1]} proposed a variational derivation through Onsager's principle of minimum energy dissipation of a new class of generalized Navier boundary conditions for two-phase flows
\(&{v}\cdot \mathbf {n}=0, \quad 2\nu (\psi ) \left( \cdot \mathbf {n}\right)_\tau + \gamma {v}_\tau = L(\psi )\nabla _\Gamma \psi &&\quad \text{on $\Sigma $},\\&\partial _t \psi + {v}_\tau \cdot \nabla _\Gamma \psi =- \beta L(\psi ),\quad &&\quad \text{on $\Sigma $},\\&\partial _\mathbf {n}\mu =0&&\quad \text{on $\Sigma $},\) 
